Books by Christopher Hollings
Drawing evidence from a range of disciplines, I study the extent to which scientists were able to... more Drawing evidence from a range of disciplines, I study the extent to which scientists were able to communicate with their counterparts on the opposite side of what became the Iron Curtain. I consider the scope that existed for personal communication between scientists, as well as for the attendance of foreign conferences, and describe how these changed over the decades. Access to publications is also dealt with: I address separately the questions of physical access, and of linguistic access. In particular, I argue that physical accessibility was generally good in both directions, but that Western scientists were afflicted by greater linguistic difficulties than their Soviet counterparts, whose major problems in accessing Western research were bureaucratic in nature.
A semigroup is a set which is closed under an associative binary operation. It therefore represe... more A semigroup is a set which is closed under an associative binary operation. It therefore represents an abstraction of the system of all self-mappings of a set, in much the same way that a group provides us with an abstract version of a system of permutations. The theory of semigroups is a relatively young branch of mathematics, with most of the major results having appeared after the Second World War. In this book, I describe the evolution of semigroup theory from its earliest origins to the establishment of a fully-fledged theory. The study of semigroups encompasses a wide range of topics, and much of it may be organised under the headings of `algebraic semigroup theory' and `topological semigroup theory'. Although these two are not mutually exclusive, the focus here is upon the development of the algebraic side of the theory.
Because of the time during which it developed, semigroup theory might be termed 'Cold War mathematics': there were thriving schools on both sides of the Iron Curtain, although the two sides were not always able to communicate with each other, or even gain access to the other's publications. One of the major themes of this book is the comparison of the approaches to the subject of mathematicians in East and West, and the study of the extent to which contact between the two sides was possible. I consider the way in which parallel developments in East and West shaped the subsequent theory of semigroups.
At the start of the book, some background is given on the abstract algebra of the early twentieth century, and on communications difficulties across the Iron Curtain. I then begin the discussion of the development of semigroup theory by considering the work of a Russian pioneer, A. K. Sushkevich, before moving on to look at certain semigroup-theoretic problems that emerged in the 1930s by analogy with similar problems for rings. Around the mid-point of the book, I describe the derivation of semigroup theory's first major structure theorems (around 1940), which might be taken as marking the beginning of a truly independent theory of semigroups. Thereafter, I indicate the ways in which the theory expanded in the following decades, both in terms of the topics considered and also through its internationalisation. The major themes of the later theory are outlined. The book concludes with an investigation of the early books, seminars and conferences on semigroups. In particular, I discuss the first international conference (Czechoslovakia, 1968), which led to the foundation of a journal devoted exclusively to semigroup research.
Papers by Christopher Hollings
Throughout E. T. Bell's writings on mathematics, both those aimed at other mathematicians and tho... more Throughout E. T. Bell's writings on mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the `arithmetization of algebra') would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell's own mathematical work. I offer an explanation for this apparent contradiction in terms of Bell's infamous penchant for mathematical `myth-making'.
During the several decades of the USSR's existence, Soviet mathematicians produced, at intervals,... more During the several decades of the USSR's existence, Soviet mathematicians produced, at intervals, a number of volumes of survey articles which provide us with a series of ‘snapshots’ of Soviet mathematics down the years. In this paper, I introduce these volumes as a resource for historians of Soviet mathematics, and consider the picture they paint of the development of abstract algebra in the USSR, paying particular attention to the aspects in which these surveys differ from later, retrospective accounts of Soviet algebra.
В течение нескольких десятилетий существования СССР советские математики с определенными интервалами выпустили несколько томов обзоров, которые дали целую серию “мгновенных снимков” советской математики за прошедшие годы. Настоящая статья имеет целью введение этих изданий в научный оборот в качестве источников по истории советской математики. В ней также исследуется создаваемая ими картина развития абстрактной алгебры в СССР. Особое внимание обращается на отличия этих обзоров от последующих ретроспективных отчетов о советской алгебре.
We consider the investigation of the embedding of semigroups in groups, a problem which spans the... more We consider the investigation of the embedding of semigroups in groups, a problem which spans the early-twentieth-century development of abstract algebra. Although this is a simple problem to state, it has proved rather harder to solve, and its apparent simplicity caused some of its would-be solvers to go awry. We begin with the analogous problem for rings, as dealt with by Ernst Steinitz, B. L. van der Waerden and Øystein Ore. After disposing of A. K. Sushkevich's erroneous contribution in this area, we present A. I. Maltsev's example of a cancellative semigroup which may not be embedded in a group, which showed for the first time that such an embedding is not possible in general. We then look at the various conditions that were derived for such an embedding to take place: the sufficient conditions of Paul Dubreil and others, and the necessary and sufficient conditions obtained by A. I. Maltsev, Vlastimil Pták and Joachim Lambek. We conclude with some comments on the place of this problem within the theory of semigroups, and also within abstract algebra more generally.
In this article, I scutinize an assertion that the Russian-Ukrainian mathematician S. O. Shatunov... more In this article, I scutinize an assertion that the Russian-Ukrainian mathematician S. O. Shatunovskii (1859–1929) should be credited with the first modern definition of a ring. Shatunovskii's claim is compared with that of Abraham Fraenkel, who defined a notion very close to the current concept of a ring in a paper of 1914.
This article provides a brief account of Soviet ideology of mathematics, beginning with a short i... more This article provides a brief account of Soviet ideology of mathematics, beginning with a short introduction to the underlying philosophy of dialectical materialism, and then examining in turn the three distinct ideological phases identified by Alexander Vucinich: before, during and after Stalin's period in power.
The Ehremann--Schein--Nambooripad Theorem expresses the fundamental connection between the notion... more The Ehremann--Schein--Nambooripad Theorem expresses the fundamental connection between the notions of inverse semigroups and inductive groupoids, which exists because these concepts provide two distinct approaches to the study of one-one partial transformations. In the case of arbitrary partial transformations, the analogous two approaches are provided by restriction semigroups and inductive categories, the former being generalisations of inverse semigroups, and the latter of inductive groupoids. There is indeed also a generalisation of the Ehremann--Schein--Nambooripad Theorem which encapsulates the connection between these two more general objects. In this article, we will explore the origins of these theorems, and survey the basic theory surrounding them.
The Mathematical Gazette, Nov 2012
We investigate two ciphers, supposedly used by Augustus, which appear in Robert Graves' I, Claudi... more We investigate two ciphers, supposedly used by Augustus, which appear in Robert Graves' I, Claudius. We focus in particular on the more complicated (and therefore more secure) of the two, the so-called cipher extraordinary, and relate it to another, better-known cipher.
Journal of the London …
We give a complete description of Green's D relation for the multiplicative semigroup of all n × ... more We give a complete description of Green's D relation for the multiplicative semigroup of all n × n tropical matrices. Our main tool is a new variant on the duality between the row and column space of a tropical matrix (studied by Cohen, Gaubert and Quadrat and separately by Develin and Sturmfels). Unlike the existing duality theorems, our version admits a converse, and hence gives a necessary and sufficient condition for two tropical convex sets to be the row and column space of a matrix. We also show that the matrix duality map induces an isometry (with respect to the Hilbert projective metric) between the projective row space and projective column space of any tropical matrix, and establish some foundational results about Green's other relations.
One of the major issues facing mathematics at present is the demand for immediate applications. A... more One of the major issues facing mathematics at present is the demand for immediate applications. Another (historical) example of such a policy may be found in Soviet ideological interference in mathematics, of which I will give a short description, before illustrating it with an account of an ideological attack on the Russian algebraist E. S. Lyapin.
International Electronic Journal of Algebra, Jan 1, 2011
A constellation is a set with a partially-defined binary operation and a unary operation satisfyi... more A constellation is a set with a partially-defined binary operation and a unary operation satisfying certain conditions, which, loosely speaking, provides a 'one-sided' analogue of a category, where we have a notion of 'domain' but not of 'range'. Upon the introduction of an ordering, we may define so-called inductive constellations. These prove to be a significant tool in the study of an important class of semigroups, termed left restriction semigroups, which arise from the study of systems of partial transformations. In this paper, we study the defining conditions for (inductive) constellations and determine that certain of the original conditions from previous papers are redundant. Having weeded out this redundancy, we show, by the construction of suitable counterexamples, that the remaining conditions are independent.
Acta Sci. Math.(to appear), Jan 1, 2010
In a previous paper, we obtained conditions on a monoid $M$ for its prefix expansion to be either... more In a previous paper, we obtained conditions on a monoid $M$ for its prefix expansion to be either a left restriction monoid (in which case $M$ must be either 'type-I' or 'type-II') or a left ample monoid ($M$ is 'type-Ia' or 'type-IIa'). In the present paper, we demonstrate that there is some redundancy in these conditions. We therefore trim down the sets of conditions and show, by construction of suitable counterexamples, that the reduced sets of conditions are independent.
Semigroup Forum, Jan 1, 2011
Inductive constellations are one-sided analogues of inductive categories which correspond to left... more Inductive constellations are one-sided analogues of inductive categories which correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids. In this paper, we define the notions of the action and partial action of an inductive constellation on a set, before introducing the Szendrei expansion of an inductive constellation, which is modelled closely on that defined by Gilbert (2005) for inductive groupoids. The main result of the paper is a theorem which uses this Szendrei expansion to link the actions and partial actions of inductive constellations, and is analogous to results previously proved by various authors for groups, monoids, and other objects.
Semigroup Forum, Jan 1, 2010
We extend the 'join-premorphisms' part of the Ehresmann-Schein-Nambooripad Theorem to the case of... more We extend the 'join-premorphisms' part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (1991) for the 'morphisms' part. However, it is so-called 'meet-premorphisms' which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for meet-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case.
Communications in Algebra, Jan 1, 2009
The Ehresmann-Schein-Nambooripad (ESN) Theorem, stating that the category of inverse semigroups a... more The Ehresmann-Schein-Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample and weakly $E$-ample semigroups. A semigroup in any of these classes must contain a semilattice of idempotents, but need not be regular. It is significant here that these classes are each defined by a set of conditions and their left-right duals.
Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroups we dub inductive constellations, together with the appropriate notion of ordered map, which we call inductive radiant. We note that such objects have appeared outside of semigroup theory in the work of Exel. In a subsequent article we develop a theory of partial action and expansion for inductive constellations, along the lines of that of Gilbert for inductive groupoids.
BSHM Bulletin, Jan 1, 2009
Anton Kazimirovich Suschkewitsch was a Russian mathematician who spent most of his working life a... more Anton Kazimirovich Suschkewitsch was a Russian mathematician who spent most of his working life at Kharkov State University in the Ukraine. In the 1920s, he embarked upon the first systematic study of semigroups, placing him at the very beginning of algebraic semigroup theory and, arguably, earning him the title of the world’s first semigroup theorist. Owing to the political circumstances under which he lived, however, his work failed to find a wide audience during his lifetime. We give a brief account of his life and his researches into semigroup theory.
Archive for history of exact sciences, Jan 1, 2009
In the history of mathematics, the algebraic theory of semigroups is a relative new-comer, with t... more In the history of mathematics, the algebraic theory of semigroups is a relative new-comer, with the theory proper developing only in the second half of the twentieth century. Before this, however, much groundwork was laid by researchers arriving at the study of semigroups from the directions of both group and ring theory. In this paper, we will trace some major strands in the early development of the algebraic theory of semigroups. We will begin with the aspects of the theory which were directly inspired by, and were analogous to, existing results for both groups and rings, before moving on to consider the first independent theorems on semigroups: theorems with no group or ring analogues.
European Journal of Pure and Applied Mathematics, Jan 1, 2009
Left restriction semigroups are a class of semigroups which generalise inverse semigroups and whi... more Left restriction semigroups are a class of semigroups which generalise inverse semigroups and which emerge very naturally from the study of partial transformations of a set. Consequently, they have arisen in a variety of different contexts, under a range of names. One of the various guises under which left restriction semigroups have appeared is that of weakly left $E$-ample semigroups, as studied by Fountain, Gomes, Gould and Lawson, amongst others. In the present article, we will survey the historical development of the study of left restriction semigroups, from the 'weakly left $E$-ample' perspective, and sketch out the basic aspects of their theory.
Journal of The Australian Mathematical Society, Jan 1, 2009
We introduce partial actions of weakly left $E$-ample semigroups, thus extending both the notion ... more We introduce partial actions of weakly left $E$-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left $E$-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation $\alpha\mapsto\alpha^+$, where $\alpha^+$ is the identity map on the domain of $\alpha$. We investigate the construction of ‘actions’ from such partial actions, making a connection with the $FA$-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schr\"{o}der, and by the second author, are to be extended appropriately to the case of weakly left $E$-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left $E$-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.
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Books by Christopher Hollings
Because of the time during which it developed, semigroup theory might be termed 'Cold War mathematics': there were thriving schools on both sides of the Iron Curtain, although the two sides were not always able to communicate with each other, or even gain access to the other's publications. One of the major themes of this book is the comparison of the approaches to the subject of mathematicians in East and West, and the study of the extent to which contact between the two sides was possible. I consider the way in which parallel developments in East and West shaped the subsequent theory of semigroups.
At the start of the book, some background is given on the abstract algebra of the early twentieth century, and on communications difficulties across the Iron Curtain. I then begin the discussion of the development of semigroup theory by considering the work of a Russian pioneer, A. K. Sushkevich, before moving on to look at certain semigroup-theoretic problems that emerged in the 1930s by analogy with similar problems for rings. Around the mid-point of the book, I describe the derivation of semigroup theory's first major structure theorems (around 1940), which might be taken as marking the beginning of a truly independent theory of semigroups. Thereafter, I indicate the ways in which the theory expanded in the following decades, both in terms of the topics considered and also through its internationalisation. The major themes of the later theory are outlined. The book concludes with an investigation of the early books, seminars and conferences on semigroups. In particular, I discuss the first international conference (Czechoslovakia, 1968), which led to the foundation of a journal devoted exclusively to semigroup research.
Papers by Christopher Hollings
В течение нескольких десятилетий существования СССР советские математики с определенными интервалами выпустили несколько томов обзоров, которые дали целую серию “мгновенных снимков” советской математики за прошедшие годы. Настоящая статья имеет целью введение этих изданий в научный оборот в качестве источников по истории советской математики. В ней также исследуется создаваемая ими картина развития абстрактной алгебры в СССР. Особое внимание обращается на отличия этих обзоров от последующих ретроспективных отчетов о советской алгебре.
Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroups we dub inductive constellations, together with the appropriate notion of ordered map, which we call inductive radiant. We note that such objects have appeared outside of semigroup theory in the work of Exel. In a subsequent article we develop a theory of partial action and expansion for inductive constellations, along the lines of that of Gilbert for inductive groupoids.
Because of the time during which it developed, semigroup theory might be termed 'Cold War mathematics': there were thriving schools on both sides of the Iron Curtain, although the two sides were not always able to communicate with each other, or even gain access to the other's publications. One of the major themes of this book is the comparison of the approaches to the subject of mathematicians in East and West, and the study of the extent to which contact between the two sides was possible. I consider the way in which parallel developments in East and West shaped the subsequent theory of semigroups.
At the start of the book, some background is given on the abstract algebra of the early twentieth century, and on communications difficulties across the Iron Curtain. I then begin the discussion of the development of semigroup theory by considering the work of a Russian pioneer, A. K. Sushkevich, before moving on to look at certain semigroup-theoretic problems that emerged in the 1930s by analogy with similar problems for rings. Around the mid-point of the book, I describe the derivation of semigroup theory's first major structure theorems (around 1940), which might be taken as marking the beginning of a truly independent theory of semigroups. Thereafter, I indicate the ways in which the theory expanded in the following decades, both in terms of the topics considered and also through its internationalisation. The major themes of the later theory are outlined. The book concludes with an investigation of the early books, seminars and conferences on semigroups. In particular, I discuss the first international conference (Czechoslovakia, 1968), which led to the foundation of a journal devoted exclusively to semigroup research.
В течение нескольких десятилетий существования СССР советские математики с определенными интервалами выпустили несколько томов обзоров, которые дали целую серию “мгновенных снимков” советской математики за прошедшие годы. Настоящая статья имеет целью введение этих изданий в научный оборот в качестве источников по истории советской математики. В ней также исследуется создаваемая ими картина развития абстрактной алгебры в СССР. Особое внимание обращается на отличия этих обзоров от последующих ретроспективных отчетов о советской алгебре.
Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroups we dub inductive constellations, together with the appropriate notion of ordered map, which we call inductive radiant. We note that such objects have appeared outside of semigroup theory in the work of Exel. In a subsequent article we develop a theory of partial action and expansion for inductive constellations, along the lines of that of Gilbert for inductive groupoids.